Hi yorgo32,
Thanks for coming to Apple Support Communities for help today.
You should be able to export the mail on the original Mac and then bring it over to your new Mac. This page from the larger Mail User Guide will be able to help: Import or export mailboxes in Mail on Mac.
"Mail can import email messages that have been exported in mbox format from other email apps, and import mailboxes exported from Mail on a Mac. You can export mailboxes in mbox format.
Import mailboxes
- In the Mail app on your Mac, choose File > Import Mailboxes.
- Select a source in the list, read the information that appears below the list, then click Continue.
- If you’re importing a mailbox exported from Mail or a folder in the ~/Library/Mail/ folder on another Mac, select Apple Mail. To get to the Mail folder on a Mac, click the desktop to make sure you’re in the Finder, press and hold the Option key, choose Go > Library, then select the Mail folder.
- If you’re importing messages from a Windows or UNIX computer, select “Files in mbox format,” then locate the folder containing the files.
3.Reorganize the imported messages, if you like.
The messages are in a new mailbox called Import in the Mail sidebar. You can drag folders and messages from the Import mailbox to new or existing mailboxes, then delete the Import mailbox.
Export mailboxes
- In the Mail app on your Mac, select one or more mailboxes, then choose Mailbox > Export Mailbox.
- Choose a folder or create a new folder, then click Choose.
Mail exports the mailboxes as .mbox packages. If you previously exported a mailbox, Mail doesn’t overwrite the existing .mbox file; it creates a new .mbox file, such as My Mailbox 3.mbox."
As far as iCloud Mail not syncing, have you checked iCloud.com from a computer's browser to see if your mail is syncing there? This could further isolate the issue. Also double-check the steps here: Set up iCloud Mail on all your devices.
"Set up iCloud Mail on your Mac
- On your Mac, do one of the following:
- macOS Ventura: Choose Apple menu > System Settings, click [your name] at the top of the sidebar, then click iCloud on the right.
- macOS 12 or earlier: Choose Apple menu > System Preferences, click Apple ID, then click iCloud.
2.If you want to send and receive iCloud Mail on your Mac, click iCloud Mail, then click Turn On or select iCloud Mail (or Mail).
3.If you want to keep Mail settings up to date across your devices, click or select iCloud Drive, click Options, then select Mail. The following settings stay up to date:
- Blocked senders (for all macOS devices)
- Email address suggestions (for all iOS, iPadOS, and macOS devices)
- Flag names (for all macOS devices)
- Mail rules (for all macOS devices)
- Signatures (for all macOS devices)
- Smart Mailboxes (for all macOS devices)
- VIPs (for all iOS, iPadOS, and macOS devices)
4.To view your iCloud Mail, open the Mail app. Click a mailbox or folder in the iCloud section.
For more information on how to use Mail on your Mac, see the Mail User Guide for Mac."
Have a fantastic day!